Transcript Lecture 11

Multi-wave Mixing
In this lecture a selection of phenomena based on the mixing of two or more waves to produce a
new wave with a different frequency, direction or polarization are discussed. This includes
interactions with non-optical normal modes in matter such as molecular vibrations which under
the appropriate conditions can be excited optically via nonlinear optics.
In nonlinear optics “degenerate” means that all the beams are at the same frequency  and
“non-degenerate” identifies interactions between waves of different frequency. Since the
interactions occur usually between coherent waves, the key issue is wavevector matching.
Because there is dispersion in refractive index with frequency, collinear wave-vector matched
non-degenerate interactions are not trivial to achieve, especially in bulk media. Of course, the
waves can be non-collinear to achieve wavevector conservation in which case beam overlap
reduces interaction efficiency.
Degenerate Four Wave Mixing (D4WM)
Beam Geometry and Nonlinear Polarization


 
k p1  k p 2  ks  kc  0


k s  k c


k p 2  k p1
“P1” and “p2” are counter-propagating pump waves, “s” is the input signal, “c” is the conjugate
General case for third order polarization, in frequency domain
(3)
Pi(3) ( )   0    ˆ ijkl
( ; 1 , 2 , 3 ) E j (1 )Ek (2 ) El (3 )δ (  1  2  3 )d1d2 d3
Each is the total field!
Each total field is given by
 (p1)
 *(p1)
 (p2)
 *(p2)


( ) (  i )  E ( ) (  i )  E
( ) (  i )
1 E ( ) ( - i )  E
E (i )   




2  E (s) ( ) ( - i )  E*(s) ( ) (  i )  E (c) ( ) ( - i )  E*(c) ( ) (  i ) 


Full 4-wave mixing process will contain 8x8x8 terms!
But, will need wavevector conservation in interaction → reduces # of terms
1. Assume beams are co-polarized along the x-axis and treat as scalar problem
2. E(p1) and E(p2) (pump beams) are strong, and E(s) and E(c) are weak beams
3. For the nonlinear polarization, only products containing both pump beams
and either the signal or conjugate beam are important
4. Nonlinear polarization (products of 3 fields) need p1, p2 and c or s
1
(3)
(3)
e.g.  Px(3) (s )   0{ˆ xxxx
(s ; p1 ,c , p 2 )  ˆ xxxx
(s ; p 2 ,c , p1 )
4
(3)
(3)
 ˆ xxxx
(s ; p1 , p 2 ,c )  ˆ xxxx
(s ; p 2 , p1 ,c )
 
i
k
(3)
(3)
 ˆ xxxx
(s ;c , p1 , p 2 )  ˆ xxxx
(s ;c , p 2 , p1 )}E x(p1) E x(p 2) E *x (c)e s r
Similar results for Px(3) (c ).
(1)
(3)
5. “grating model”: products of two fields create a grating  xx
,effective   xxxx E x (1 )E x ( 2 )
in space, and a third wave is deflected by the grating to form beams s or c. Initially the only 2
field products of interest are (p1 or p2) x (c
or s).
  
  
1
i
(
k

k
)

r
i
(
k
 k ) r
 [{ (1   ) (2   )}{2E (p1)E (s)e p s  2E (p1)E (c) e p s
4
 2E
 2E
  
i
(

k
p  k s ) r
(p2) (s)
E
 2E
  
(p1) *(s) i ( k p  ks )r
E
e
  
i
(

k
p  k s ) r
(p2) (c)
 2E
E
}  { (1   ) (2   )}{
  
(p1) *(c) i ( k p  ks )r
E
e
 2E
  

i
(
k
p  k s ) r
(p2) *(s)
E
 2E
  
i
(

k
p  k s ) r
(p2) *(c)
E
}  c.c.]
There are two kinds of time dependences present here corresponding
(3)
to the first two inputs (1 ,  2 ) in ˆ xxxx ( ; 1 , 2 , 3 )
1.  (1   ) ( 2   )  c.c. oscillates at 2 (requires electronic nonlinearity)
2.  (1   ) ( 2   )  c.c. DC in time
- Now form E(1)E(2) x E(3) subject to the following restrictions
1. Terms   (  1) (  2 ) can only multiply terms with  (  3 ) so that the
output frequency is 
2. The product of three different beams is required
3. Because the pump beams are the “strong” beams, only products with two pump beams are
kept which generate signal or conjugate beams. Assume Kleinman limit.




6  (3)
(3)
(p1) (p2) * (s) iks r
*(c) iks r
E (1 ) E (2 ) E (3 )  Px ( )   0  xxxx ( ;  , ,  )E x E x {E x e
 Ex e
}


4


 Px(3) ( )  2n 2c 02 n2|| (- ;  )E x(p1)E x(p2){E*x (s)e iks r  E*x (c)e iks r }
First term generates the conjugate and the second term the signal.
D4WM Field Solutions
Using the SVEA in undepleted
pump beams approximation

d (c)
E ( z ' ,  )  in 0 n2 E (p1) ( )E (p2) ( )E *(s) ( z ' ,  )
dz '
d (s)
E ( z ' ,  )  in 0 n2 E (p1) ( )E (p2) ( )E *(c) ( z ' ,  )
dz '
Simplifying in the undepleted pump approximation
d (c)
1
1
E ( z ' )  i
E *(s) ( z ' )
 n 0 n2 E (p1) ( )E (p2) ( )
dz '
 4WM
 4WM
d (s)
1
E ( z' )  i
E *(c) ( z ' )
dz '
 4WM
Applying the boundary conditions: E (c) ( L' ,  )  0
E (s) (0,  )  0
 L'
  4WM
 Output of conjugate beam: E (c) (0,  )  iE *(s) (0,  ) tan 



2  L' 



tan
(s)
2
| E (0,  ) |
  4WM 
| E (s) ( L' ,  ) |2
2  L' 


" Transmissivity"  (s)

sec
2
| E (0,  ) |
  4WM 
" Reflectivity" 
| E (c) (0,  ) | 2
R>1 ? YES! Photons come out of pump beams!
Need to include pump depletion.
L'
 4WM

 Can get gain on both beams!

 " oscillation" and R   and T   Unphysical, need pump depletion
2
Manley Rowe Relation
E *(c) ( z ' ) x
1
d (c)
E *(s) ( z ' ) xE *(c) ( z ' )
E ( z ' )  i
 4WM
dz '
E *(s) ( z ' ) x
1
d (s)
E *(c) ( z ' ) xE *(s) ( z ' )
E ( z' )  i
 4WM
dz '
d (s)
d (c)

I ( z' )  
I ( z' )
dz '
dz '
Since signal travels along +z, and the conjugate travels
along –z, both beams grow together at expense of the
pump beams.
Can be shown easily when z  z and allowing pump beam depletion
d (p1)
E
( z ,  )  in 0 n2 E (s) ( z ,  )E (c) ( z ,  )E *(p2) ( z ,  )
dz
d (p2)
E
( z ,  )  in 0 n2 E (s) ( z ,  )E (c) ( z ,  )E *(p1) ( z ,  )
dz
Note that the p1 and s, and the p2 and c beams travel in the same direction

d ( p1)
d
I ( z )  I (s ) ( z )
dz
dz
Pump beam #1 depletes
and
d ( p 2)
d
I
( z )   I (c) ( z )
dz
dz
Signal beam grows Pump beam #2 depletes Conjugate beam grows
Wavevector Mismatch
What if pump beams are misaligned, i.e. not exactly parallel?
 

 
k  kp1  kp 2  ks  kc Assume that z and z are essentially coincident
d (c)
1
E ( z )  i
E *(s) ( z )e ikz
dz
 4WM
d (s)
1
E ( z)  i
E *(c) ( z )e ikz
dz
 4WM
2
Form of solutions, subject to the usual boundary conditions with  2  k 2 / 4   4WM

2
 R max ( L   / 2)  
 k 4WM
sin 2 ( L)
R
k 4WM 2
cos 2 ( L)  [
]
2
Linear Absorption
2
absorption of all 4 beams, no pump depletion approximation to signal
and conjugate used

 z
(p1)
(p1)
E (z)  E (0)e 2 ;





E
(p2)
( z)  E
(p2)
(0)e 2
( z L)
d (s)
i

d (c)
i

E ( z) 
E *(c) ( z )  E (s) ( z );
E ( z)  
E *(s) ( z )  E (c) ( z )
dz
 4WM
2
dz
 4WM
2
Redefine
1
 4WM
 
2
  n 0 n2E (p1) (0)E (p2) ( L)e
2
 4WM
 [ / 2]
2

L
2
R 
sin 2 ( L)

 24WM [  cos( L)  { } sin( L)] 2
2
Complex ̂
(3)
But
̂ (3) is in general a complex quantity, i.e.  (3)  i (3)
index change absorption change
R
2

2|| (  ;  ) ( p1)
2 2
( p 2)
4(k vac L) {n2|| ( ;  ) 
}
I
(
0
)
I
( L)
2
4k vac
 Both the real (n2) and imaginary (2) parts of ̂ (3) contribute to D4WM signal
Three Wave Mixing
y
z
x
Assume a thin isotropic medium. Co-polarized beams,
x-polarized with small angles between input beams


k p1  (k y , k z )  k ( sin  , cos  ); k p 2  k (sin  , cos  )
[E (p1) ] 2 E *(p2) and [E (p 2) ] 2 E *(p1)
Look at terms
P (s1) ( z )  [n 0 ]2 cn2|| (- ;  )[E (p 2) ]2 E
P (s2) ( z )  [n 0 ]2 cn2|| (- ;  )[E (p1) ]2 E




i
(
2
k

k
*(p1)
p2
p1 )r
e



i
(
2
k

k
*(p 2)
p1
p 2 )r
e


0 2 ( s1, s 2) i (2 k p 2, p1  k p1, p 2  k s1, s 2 ) z
E ( s1, s 2) (z)
SVEA 
i
P
e
z
2k s1, s 2




d (s1)
i ( 2 kp 2  kp1  ks1 )r
E ( z )  i s1e
dz


 
d (s2)
i ( 2 kp1  k p 2  ks 2 )r
E ( z )  i s 2 e
dz
n 0 n2


 s1 
[E
] E
2k p 2  k p1  k (3 sin  , cos  )
2

n 0 n2 (p1) 2 *(p 2) 
 s2 
[E ] E
2k p1  k p 2  k (3 sin  , cos  )
2
(p 2) 2
*(p1)
Assuming that the beams are much wider in the x-y plane than a wavelength, wavevector is
conserved in the x-y plane. For the signal field which must be a solution to the wave equation,
9
9
k z2  k 2  k y2  k 2  9k 2 sin 2   k 2 [1  9 sin 2  ]  k z  k (1  sin 2  )  k (1   2 )
2
2
9 2
 k z  2k z , p 2  k z , p1  k z ,s1  k z , p1  k z , p 2  k z ,s 2  k (cos   1   )  4k 2
2
k z L ( p 2)
I
(0)]2 I ( p1) (0),
2
k L
I (s 2) ( L)  [k vac Ln2|| ( ;  )sinc 2 z I ( p1) (0)]2 I ( p 2) (0).
2
Integrating the SVEA equations : I (s1) ( L)  [k vac Ln2|| ( ;  )sinc 2
E (s1) and E (s2) can interact with the pump beams E (p1) and E (p2)
again to produce more output beams etc.
Called the Raman Nath limit of the interaction
Non-degenerate Wave Mixing
In the most general non-degenerate case with frequency inputs, 1 , 2 , 3 and 4 in which 1
and  2 are the pump beams, then for frequency 12  3  4 and wavevector conservation,






k (1)
k ( 2 )
1
k (1)  k ( 2 )  k ( 3)  k ( 4 )  {1n(1 )
 2 n(2 )
c
| k (1) |
| k ( 2 ) |


k ( 3)
k ( 4 )
 3n(3 )
 4 n(4 )
} 0
| k ( 3) |
| k ( 4 ) |
A frequent case is one pump beam from which two photons at a time are used to generate two
signal beams at frequencies above and below the pump frequency which, for efficiency, requires
21  3  4 ;
k  21n(1 )  3n(3 )  4 n(4 )  0
The fields are written as:
 (1) 1 
 (3) 1  
 ( 4) 1 
i ( k3 z 3t )
i ( k1z 1t )
E  E (1 )e
 c.c. ; E  E (r ;3 )e
 c.c. ; E  E (3 )ei ( k4 z 4t )  c.c.
2
2
2
Assuming
(1) co-polarized beams,
(2) the Kerr effect in the non-resonant regime,
(3) cross-NLR due to the pump beam only,
(4) and a weak signal (3) input,
the signal and idler (conjugate) beam
nonlinear polarizations are
P ( NL ) (3, 4 )  2
6 ~ (3)
 xxxx (3,4 ; 1,1, 3,4 ) I1E (3,4 )
4n1c
3
i[ k  2 n2|| ( 1;1 ) k vac (1 ) I1 ] z
(3)
  0 ~xxxx
(3, 4 ; 1 , 1 ,4,3 )[E (1 )]2 E * (4,3 )e
}
4
n
Defining :  3, 4  2k vac (3, 4 ) 1 n2|| (1; 1 ) I1   k  2k vac (1 )n2|| (1; 1 ) I1
n3, 4
n1
n2|| (1; 1 ) I1e 2i (1 ) ,
n3, 4
d
and substituting into the SVEA :
E (3, 4 , z )  i 3, 4E (3, 4 , z )  i3, 4E * (4,3 , z )eiz
dz
Using the substitutions : E (3 , z )  B3 ( z )ei 3 z
E (4 , z )  B4 ( z )ei 4 z ,
3, 4  k vac (3, 4 )

d2
d
B



B

i
[


(



)]
B3 ;
3 4 3
3
4
2 3
dz
dz
d2
dz 2
B4  34 B4  i[  ( 3   4 )]
For solutions of the form : B3, 4  B3, 4e  z  B3, 4e  z
i (   3   4 ) 1
(   3   4 )

 [  ( 3   4 )]2  434  i
 g.
2
2
2
The signal and idler grow exponentially for 434  [  ( 3   4 )]2 when g is real!!
 
For imaginary g, the solutions are oscillatory
For the boundary conditions E (4 ,0)  0, E (3 ,0)  0
d
B4
dz
E (3 , z )  E (3 ,0)[cosh( zg )  i
  ( 3   4 )
2g
i
sinh( gz )]e
  ( 3  4 )
2
z
,
  ( 3  4 )
z
4 *
2
E (4 , z )  i E (3 ,0) sinh( gz )e
.
g
A number of simplifications can be made which give insights into the conditions for gain.
2
Expanding Δk around k (1 )  2k (1 )  k (3 )  k (4 )  k 2 (1 ) for small
  4  1  1  3 . The sign of Δk is negative in the normal dispersion region and positive
in the anomalous dispersion region. The condition for gain can now be written as
n
[2 1
k vac (3 )k vac (4 )n2|| (1;1 ) I1 ]2
n3n4
i
 {k  n2|| (1;1 ) I1[2k vac (1 ) 
n1
n
k vac (3 )  1 k vac (3 )]}2  0
n3
n4
After some tedious manipulations valid for small Δ, the condition for gain becomes
1
1
[2k vac (1 )n2|| (1;1 ) I1  k 2  2 ]  [2k vac (1 )n2|| (1;1 ) I1  k 2  2 ]  0.
2
2
Gain occurs for both signs of the GVD and the nonlinearity provided that the intensity
exceeds the threshold value
1
2k vac (1 ) | n2|| (1;1 ) | I1 | k | | k 2 |  2 .
2
This means that the cross-phase nonlinear refraction due to the pump beam must exceed the
index detuning from the resonance for gain to occur.
Nonlinear Raman Spectroscopy
Usually refers to the nonlinear optical excitation of vibrational or rotational modes. A minimum
of two unique input beams are mixed together to produce the normal mode at the sum or
difference frequency. Although any Raman-active mode will work, vibrational modes typically
are very active in modulating the polarizability.
Note: Must include dissipative loss of the normal modes in Manley-Rowe relations
Degenerate Two Photon Vibrational Resonance
a
a
Optical coupling between two vibrational levels (inside the vibrational
manifold of the electronic ground state)
 ijk
 L

ij  polarizabi lity tenso r  αij  qk
qk 0 

q
k


Vibrational amplitude
e.g. the symmetric breathing mode in a methane
molecule (CH4)
H
H
H
C
C
H
H
H
H
H
 
 
1 
Einc (r , t )  E (a )ei ( ka r at )  c.c.
2
 NL

From previous discussions  p  q
q
  2 v1q  v2 q  1 
classical mechanics  q
2m q

q  0  (NL )  (a ) E (a )
(1)
(1)
 
q 0 [  (a )] E  E
(1)
2

2


1 E 2 (a ) 
2i ( k a r a t ) | E (a ) | 
( 2)
 q (r , t )  [

q 0 e
q  0  c.c.]
2 4m D(2a ) q
4m D(0) q
Note: The molecular vibrations are not only driven
in time due to the field mixing, but also into a
spatial pattern for the first term
q
/ka
z
N
For resonant two photon absorption  P NL (a ) 
8m
(3)
(1)
3 (1)
 NL  2a  f  [ f (a )] f (2a )
a N  
d
SVEA  E (a )  i
dz
16na m  0c  q
 
 q

2
 (3) | E (a ) |2 E (a )
q 0  
v2  4a2  4ia v-1

D(2a)
2
 (3) | E (a ) |2 E (a ) v  2a  i v-1
q 0  
4a
( v  2a ) 2  ( v-1 ) 2

Real part gives 2 photon absorption; Imaginary part gives index change
N v-1
d
For 2PA near resonance  E (a )  
dz
32na2 m  02c 2
 
 q

2
 (3)
I (a )E (a )

q 0 
( v  2a ) 2  ( v-1 ) 2

 
d
N
 I (a )   2|| (a ; a ) I 2 (a )   2|| (a ; a ) max 

dz
16na2 m  02c 2 v-1  q
d
N
For associated n2|| ( ;  ) :
E (a )  i
dz
32na2 m  02c 2
 
 q

2
 (3)
v  2a

I (a )E (a )
q 0 
2
-1 2
(


2

)

(

)

v
a
v
d
E (a )  ik vac n2 (-a ; a ) I (a )E (a )
dz
 
 n2|| (a ; a ) 

32na2 02 m ca  q
N
v  2a  n2|| ( ;  )  0
2
 (3)
q 0  

2
 (3)
v  2a

q 0 
( v  2a ) 2  ( v-1 ) 2

2a  v  n2|| ( ;  )  0
Nonlinear Raman Spectroscopy
Nonlinear process drives the vibration at the difference frequency a-b between input fields
 
 


1 
i ( k a r a t )
i ( kb r bt )
Einc   E (a )e
 c.c.  E (b )e
 c.c.
2

I
RIKES – Raman Induced Kerr Effect Spectroscopy ( Raman Induced Birefringence)

E(a ) x  i - polarized strong field Makes medium birefringent for beam “b” and changes

E(b ) y  j - polarized weak field transmission of medium “b”
II
CARS – Coherent Anti-Stokes Raman Spectroscopy

E(b ) - strong field

E(a ) - strong field
New fields generated at 2a-b and 2b-a
(only one can be phase-matched at a time)
ijn
ijn
 L

NL
(1)
(1)
N
RIKES pi  ij  qn
qn 0  Eloc, j  Pi
q n 0 qn  ( NL )  (b ) E j
qn
qn


1  ijn
(1)
(1)
classical mechanics : qn  2 v1q n  v2 qn 
qn 0  (b )  (a ) Ei (a ) E j (b )
m qn

1  ijn
 qn ( r , t ) 
4m qn
qn 0 
(1)
(a ) 
(1)
(b )
Ei* (a )E j (b )
D* (a  b )
e
  
i[( kb  ka )r (b a )t ]
 c.c.
Since  NL  b , define  (3)  [  (1) (a )  (1) (b )]2 , the nonlinear polarization at b is
N  jin
NL 
 Pj (r , t ) 
8m qn
q n 0 x
*
 ijn
qn
qn 0 
2
|
E
(

)
|
E j (b )
i
a
(3)
D (a  b )
*
e
 
i[ kb r bt ]
 c.c
SVEA 
 jin
d
Nb
E j (b )  i
[
2 2
dz
8nbna m  0 c v qn
1.
2.
3.
4.
q n 0
ijn
qn
qn  0 ]
( 3)
v  [a  b ]  i v-1
I ( )E j (b )
2
-1 2 i a
( v  [a  b ])  ( v )
Index change produced at frequency b by beam of frequency a
Nonlinear gain or loss induced in beam “b” by beam “a”
One photon from beam “a” breaks up into a “b” photon and an optical phonon
Propagation direction of beam “a” is arbitrary, only polarization important!
Imaginary part contribution → to nonlinear refractive index coefficient

d
E j (b )  i b n2 (-b ; a ) I i (a )E j (b )
dz
c
 jin
 ijn
N
 n2 (b ; a ) 
[
q 0
qn
8nb na m  02cv qn n
(3)
qn 0 ]
v  [a  b ]
( v  [a  b ]) 2  ( v-1 ) 2
Real part → contribution to nonlinear gain (or loss)
 jin
Nb v-1
d
E j (b ) 
[
dz
8nb na m  02c 2 v qn
 2 (b ; a ) 
Solving 

Nb v-1
4nb na m  02c 2 v
[
q n 0
 jin
qn
 ijn
qn
(3)
]

qn 0
2 (3)
qn 0 ] 
I i (a )E j (b )
( v  [a  b ]) 2  ( v-1 ) 2
1
( v  [a  b ]) 2  ( v-1 ) 2
I (b , L)
 e 2 ( b ;a ) I (a ) L  1   2  (b ; a ) I (a ) L
I (b ,0)
I (b , L)  I (b ,0)
  2  (b ; a ) I (a ) L
I (b ,0)
Modulating the intensity of beam “a” modulates the transmission of beam “b”. Varying a - b
through v gives a resonance in the transmission! Assumed was a crystal. If medium is random,
need to work in both lab and molecule frames of reference and then average over all orientations.
CARS
Coherent Anti-Stokes Raman Scattering (Spectroscopy)
For simplicity, assume two input co-polarized beams, b>a.



NL
Looking for P (r ,2b  a )  Q (r , b  a )E( r , b )  ei(2b a )t
2
*
N  iin
NL
2 (3) Ei (b )Ei (a )
 Pi (2b  a ) 
|
qn 0 | 
4m qn
D(b  a )
N c

d
Ei (2b  a )  i
| iin
dz
8n(c )c 0 m qn
Ei2 (b )Ei* (a ) ikr
e
qn 0 |  (a )[  (b )]  ( 2b  a )
D(b  a )
2
(1)
(1)
2 (1)
1
c  2b  a is written as Ei (c )ei[ k (c ) z ct ]  c.c.
2
This process requires wavevector
 matching
 to be efficient,
k (b ) 
k  2kb  ka  kc

k (b )
k (a )
Cannot get collinear wavevector matching
Field at

k (c )
because of index dispersion in the visible
na  nc  2nb  k (c )  k (a )  2k (b )
since c >> b - a then angles are small relative to z-axis
I ( L,  c ) 
N c2 L2
64nc nb2 na c 4 04 m 2 v2
2
|
 iin
qn
L
sinc 2 (k ) I 2 (b ) I (a )
4
(3) 2
2
qn 0 | [  ]
[ v  (b  a )]2  ( v-1 ) 2
When (b  a ) is tuned through v , resonant
enhancement in the signal occurs. Monolayer
sensitivity has been demonstrated. There is also a
“background” contribution due to electronic
(3)
transitions via:  xxxx
([2b  a ]; b ,a , b )
For comparable contributions of
background and resonance terms
Can also have CSRS (different
wavevector matching conditions)
Signal appears at (2a  b )
Differences between RIKES
Automatically wave-vector matched in
isotropic medium. No new wave appears.
and
CARS
Have resonance
at v | a  b |
Requires
wave-vector
matching